This tutorial is a static non-editable version. You can launch an
interactive, editable version without installing any local files
using the Binder service (although note that at some times this
may be slow or fail to open):

All of the basic SI units can be used (volt, amp, etc.) along with all
the standard prefixes (m=milli, p=pico, etc.), as well as a few special
abbreviations like mV for millivolt, pF for picofarad, etc.

1000*amp

\[1.0\,\mathrm{k}\,\mathrm{A}\]

1e6*volt

\[1.0\,\mathrm{M}\,\mathrm{V}\]

1000*namp

\[1.0\,\mathrm{\mu}\,\mathrm{A}\]

Also note that combinations of units with work as expected:

10*nA*5*Mohm

\[50.0\,\mathrm{m}\,\mathrm{V}\]

And if you try to do something wrong like adding amps and volts, what
happens?

If you haven’t see an error message in Python before that can look a bit
overwhelming, but it’s actually quite simple and it’s important to know
how to read these because you’ll probably see them quite often.

You should start at the bottom and work up. The last line gives the
error type DimensionMismatchError along with a more specific message
(in this case, you were trying to add together two quantities with
different SI units, which is impossible).

Working upwards, each of the sections starts with a filename (e.g.
C:\Users\Dan\...) with possibly the name of a function, and then a
few lines surrounding the line where the error occurred (which is
identified with an arrow).

The last of these sections shows the place in the function where the
error actually happened. The section above it shows the function that
called that function, and so on until the first section will be the
script that you actually run. This sequence of sections is called a
traceback, and is helpful in debugging.

If you see a traceback, what you want to do is start at the bottom and
scan up the sections until you find your own file because that’s most
likely where the problem is. (Of course, your code might be correct and
Brian may have a bug in which case, please let us know on the email
support list.)

Let’s start by defining a simple neuron model. In Brian, all models are
defined by systems of differential equations. Here’s a simple example of
what that looks like:

tau=10*mseqs='''dv/dt = (1-v)/tau : 1'''

In Python, the notation ''' is used to begin and end a multi-line
string. So the equations are just a string with one line per equation.
The equations are formatted with standard mathematical notation, with
one addition. At the end of a line you write :unit where unit
is the SI unit of that variable.

Now let’s use this definition to create a neuron.

G=NeuronGroup(1,eqs)

In Brian, you only create groups of neurons, using the class
NeuronGroup. The first two arguments when you create one of these
objects are the number of neurons (in this case, 1) and the defining
differential equations.

Let’s see what happens if we didn’t put the variable tau in the
equation:

An error is raised, but why? The reason is that the differential
equation is now dimensionally inconsistent. The left hand side dv/dt
has units of 1/second but the right hand side 1-v is
dimensionless. People often find this behaviour of Brian confusing
because this sort of equation is very common in mathematics. However,
for quantities with physical dimensions it is incorrect because the
results would change depending on the unit you measured it in. For time,
if you measured it in seconds the same equation would behave differently
to how it would if you measured time in milliseconds. To avoid this, we
insist that you always specify dimensionally consistent equations.

Now let’s go back to the good equations and actually run the simulation.

First off, ignore that start_scope() at the top of the cell. You’ll
see that in each cell in this tutorial where we run a simulation. All it
does is make sure that any Brian objects created before the function is
called aren’t included in the next run of the simulation.

Secondly, you’ll see that there is an “INFO” message about not
specifying the numerical integration method. This is harmless and just
to let you know what method we chose, but we’ll fix it in the next cell
by specifying the method explicitly.

So, what has happened here? Well, the command run(100*ms) runs the
simulation for 100 ms. We can see that this has worked by printing the
value of the variable v before and after the simulation.

By default, all variables start with the value 0. Since the differential
equation is dv/dt=(1-v)/tau we would expect after a while that v
would tend towards the value 1, which is just what we see. Specifically,
we’d expect v to have the value 1-exp(-t/tau). Let’s see if
that’s right.

print('Expected value of v = %s'%(1-exp(-100*ms/tau)))

Expectedvalueofv=0.99995460007

Good news, the simulation gives the value we’d expect!

Now let’s take a look at a graph of how the variable v evolves over
time.

This time we only ran the simulation for 30 ms so that we can see the
behaviour better. It looks like it’s behaving as expected, but let’s
just check that analytically by plotting the expected behaviour on top.

As you can see, the blue (Brian) and dashed red (analytic solution)
lines coincide.

In this example, we used the object StateMonitor object. This is
used to record the values of a neuron variable while the simulation
runs. The first two arguments are the group to record from, and the
variable you want to record from. We also specify record=0. This
means that we record all values for neuron 0. We have to specify which
neurons we want to record because in large simulations with many neurons
it usually uses up too much RAM to record the values of all neurons.

Now try modifying the equations and parameters and see what happens in
the cell below.

We’ve added two new keywords to the NeuronGroup declaration:
threshold='v>0.8' and reset='v=0'. What this means is that
when v>1 we fire a spike, and immediately reset v=0 after the
spike. We can put any expression and series of statements as these
strings.

As you can see, at the beginning the behaviour is the same as before
until v crosses the threshold v>0.8 at which point you see it
reset to 0. You can’t see it in this figure, but internally Brian has
registered this event as a spike. Let’s have a look at that.

The SpikeMonitor object takes the group whose spikes you want to
record as its argument and stores the spike times in the variable t.
Let’s plot those spikes on top of the other figure to see that it’s
getting it right.

A common feature of neuron models is refractoriness. This means that
after the neuron fires a spike it becomes refractory for a certain
duration and cannot fire another spike until this period is over. Here’s
how we do that in Brian.

As you can see in this figure, after the first spike, v stays at 0
for around 5 ms before it resumes its normal behaviour. To do this,
we’ve done two things. Firstly, we’ve added the keyword
refractory=5*ms to the NeuronGroup declaration. On its own, this
only means that the neuron cannot spike in this period (see below), but
doesn’t change how v behaves. In order to make v stay constant
during the refractory period, we have to add (unlessrefractory) to
the end of the definition of v in the differential equations. What
this means is that the differential equation determines the behaviour of
v unless it’s refractory in which case it is switched off.

Here’s what would happen if we didn’t include (unlessrefractory).
Note that we’ve also decreased the value of tau and increased the
length of the refractory period to make the behaviour clearer.

So what’s going on here? The behaviour for the first spike is the same:
v rises to 0.8 and then the neuron fires a spike at time 8 ms before
immediately resetting to 0. Since the refractory period is now 15 ms
this means that the neuron won’t be able to spike again until time 8 +
15 = 23 ms. Immediately after the first spike, the value of v now
instantly starts to rise because we didn’t specify
(unlessrefractory) in the definition of dv/dt. However, once it
reaches the value 0.8 (the dashed green line) at time roughly 8 ms it
doesn’t fire a spike even though the threshold is v>0.8. This is
because the neuron is still refractory until time 23 ms, at which point
it fires a spike.

Note that you can do more complicated and interesting things with
refractoriness. See the full documentation for more details about how it
works.

This shows a few changes. Firstly, we’ve got a new variable N
determining the number of neurons. Secondly, we added the statement
G.v='rand()' before the run. What this does is initialise each
neuron with a different uniform random value between 0 and 1. We’ve done
this just so each neuron will do something a bit different. The other
big change is how we plot the data in the end.

As well as the variable spikemon.t with the times of all the spikes,
we’ve also used the variable spikemon.i which gives the
corresponding neuron index for each spike, and plotted a single black
dot with time on the x-axis and neuron index on the y-value. This is the
standard “raster plot” used in neuroscience.

The line v0:1 declares a new per-neuron parameter v0 with
units 1 (i.e. dimensionless).

The line G.v0='i*v0_max/(N-1)' initialises the value of v0 for
each neuron varying from 0 up to v0_max. The symbol i when it
appears in strings like this refers to the neuron index.

So in this example, we’re driving the neuron towards the value v0
exponentially, but we fire spikes when v crosses v>1 it fires a
spike and resets. The effect is that the rate at which it fires spikes
will be related to the value of v0. For v0<1 it will never fire
a spike, and as v0 gets larger it will fire spikes at a higher rate.
The right hand plot shows the firing rate as a function of the value of
v0. This is the I-f curve of this neuron model.

Note that in the plot we’ve used the count variable of the
SpikeMonitor: this is an array of the number of spikes each neuron
in the group fired. Dividing this by the duration of the run gives the
firing rate.

Often when making models of neurons, we include a random element to
model the effect of various forms of neural noise. In Brian, we can do
this by using the symbol xi in differential equations. Strictly
speaking, this symbol is a “stochastic differential” but you can sort of
thinking of it as just a Gaussian random variable with mean 0 and
standard deviation 1. We do have to take into account the way stochastic
differentials scale with time, which is why we multiply it by
tau**-0.5 in the equations below (see a textbook on stochastic
differential equations for more details).

That’s the same figure as in the previous section but with some noise
added. Note how the curve has changed shape: instead of a sharp jump
from firing at rate 0 to firing at a positive rate, it now increases in
a sigmoidal fashion. This is because no matter how small the driving
force the randomness may cause it to fire a spike.

That’s the end of this part of the tutorial. The cell below has another
example. See if you can work out what it is doing and why. Try adding a
StateMonitor to record the values of the variables for one of the
neurons to help you understand it.

You could also try out the things you’ve learned in this cell.

Once you’re done with that you can move on to the next tutorial on
Synapses.